5.1 Relative support for competing propositions
Suppose that there are two competing propositions, \(H_p\) for the prosecution and \(H_d\) for the defence. There is observed evidence \(E\), and the expert needs to determine whether \(E\) was more likely assuming \(H_p\) or \(H_d\), since this will assist the court in assessing the truth of \(H_p\) or \(H_d\). If \(E\) is more likely assuming \(H_p\) were true compared to if \(H_d\) were true, then the evidence \(E\) provides more support for \(H_p\) than \(H_d\), and vice versa if \(E\) is more likely assuming \(H_d\) to be true compared to \(H_p\). The likelihood ratio (LR) quantifies the magnitude of this support.
The LR is the relative size of the probability of observing \(E\) conditioned on \(H_p\), compared to the probability of observing \(E\) conditioned on \(H_d\). Mathematically, this is equivalent to dividing the probability of observing \(E\) conditioned on \(H_p\), by the probability of observing \(E\) conditioned on \(H_d\). As a formula it is written as \[\text{LR}=\frac{\text{probability of }E\text{ assuming }H_p \text{ is true}}{\text{probability of }E\text{ assuming }H_d \text{ is true}}.\]
By assuming each of the competing propositions to be true, we can see how much more likely \(E\) was to occur in the prosecution’s version of events compared to the defence’s.
Each term in the LR is technically a likelihood, whose value is a number greater than 0. We use the word ‘probability’ in the equation above and in our general discussion of LRs to avoid a technical detail that is outside the scope of this book. The LR itself is a value greater than 0. Values of the LR which are greater than 1 indicate relative support in favour of \(H_p\) compared to \(H_d\), since it means that the probability of \(E\) assuming that \(H_p\) is true is greater than the probability of \(E\) assuming \(H_d\) is true. Values of the LR which are less than 1 indicate relative support in favour of \(H_d\) compared to \(H_p\), since it means that the probability of \(E\) assuming that \(H_d\) is true is greater than the probability of \(E\) assuming \(H_p\) is true. Values of the LR which are equal to 1 indicate that evidence \(E\) provides equal support for \(H_p\) and \(H_d\). This is shown in Table 5.1.
| LR | Meaning |
|---|---|
| less than 1 | Evidence \(E\) provides more support for \(H_d\) compared to \(H_p\) |
| equal to 1 | Evidence \(E\) provides equal support for \(H_p\) and \(H_d\) |
| greater than 1 | Evidence \(E\) provides more support for \(H_p\) compared to \(H_d\) |